Effects of head loss, surface tension, viscosity and density ratio on the Kelvin–Helmholtz instability in different types of pipelines
Introduction
Hydrodynamic instability is an important issue in fluid mechanics and usually occurs at the interface of different fluids. There are three typical interface instabilities known as Rayleigh–Taylor instability (RTI), Richtmyer–Meshkov instability (RMI) and Kelvin–Helmholtz instability (KHI) [1], [2]. The RTI takes place at a fluid interface where the density gradient and the acceleration are oppositely directed [3], and the RMI arises when a shock wave passes through an interface between two fluids [4]. Both RTI and RMI are in the direction perpendicular to the interface, while KHI is induced by a parallel shear to the interface [5]. These instabilities play a key role in many fields, such as astrophysics [6], [7], plasmas [8], [9], [10], superfluid [11], [12], [13], magnetic fluid [14], [15], [16], [17] and inertial confinement fusion (ICF) [18], [19], [20], [21].
Understanding the dynamics of any instability is important not only for its own sake, but also in the research of other hydrodynamic instabilities. Indeed, KHI aggravates the development of non-linear RTI and RMI and is the key to the evolution of the mushroom structures around interface [2], [22], and it plays a critical role in the time-dependent transition to turbulence of flows driven by RT and RM instabilities [23], [24].
The linear growth of KHI has been well established [25], but the late-time dynamics is still an area of active research [2]. During the past decades, the effects of many factors on the evolution of KHI have been investigated extensively. Such factors include viscosity [26], [27], [28], [29], [30], [31], [32], surface tension [29], [30], [33], density gradient/ratio [30], [31], [33], [34], [35], [36] and so forth. The viscosity and surface tension were found to have a stabilizing effect on the KHI [26], [27], [28], [29], [30], [31], [32], [33]. The density gradient/ratio effect was shown to aggravate the KHI [30], [31], [34], [35], [36], while the opposite conclusion was drawn in Ref. [33].
Now it is necessary to comprehensively analyze and fully understand the combined effects of different factors in different types of pipelines on the growth of KHI. Also, to our knowledge, the research on the effect of head loss on the KHI is rare. In fact, in any pipeline with viscous fluid flow, the head loss, surface tension, viscosity and fluid density ratio are all factors influencing the evolution of KHI. Among these factors, the head loss is even more important [37]. It is an energy loss [38] and occurs mainly when the fluid velocity and the cross-section of pipeline change [39], [40]. The calculation of head loss is a very important problem in engineering, and its numerical value is directly related to the determination of power equipment capacity. The impact of head loss in the straight pipeline on RTI and RMI has been investigated and these instabilities were found to disappear when the head loss coefficient tends to 0.5 [41]. The energy losses caused by different head loss coefficients are also different [42]. In the KHI, when the cross-section of pipeline and the flowing direction of fluid change, the kinetic energy and potential energy of each particle inside the fluid are converted to each other, and the energy loss appears. By calculating the influence of head loss on the instability, we can design the pipe effectively to reduce the head loss in fluid flow.
The scope of this work is to investigate analytically the combined effects of head loss, surface tension, viscosity and density ratio on the growth of KHI in two typical pipelines, i.e., straight pipeline with different cross-sections and bend pipeline. We will find in the straight pipeline with different cross-sections that, when the effect of head loss is combined, the density ratio effect enforces the KHI, contrary to the result of Ref. [33], but consistent with the recent results [34], [35].
Section snippets
Models
For a simplification, a one-dimensional (1D) two-fluid model is employed and the following two typical pipelines are considered.
Straight pipeline with different cross-sections
(a) Effect of head loss
In order to show clearly the effect of head loss on the relative tangential velocity of fluid, we take the disturbance amplitude and the pipe diameter . Fig. 3 presents the wave number dependence of the relative tangential velocity for different head loss coefficients. Here , , the other parameters are fixed as follows [27]: , , , , . The velocity is found to decrease first and then
Conclusion
We have investigated the combined effects of head loss, surface tension, viscosity and density ratio on the KHI. The dynamic governing equations for the upper and lower fluids flowing in the straight pipeline with different cross-sections and bend pipeline were resolved analytically. We found in the straight pipeline with different cross-sections that the relative tangential velocity is depressed by the head loss, viscosity and density ratio of upper and lower fluids but enhanced by the surface
CRediT authorship contribution statement
X.C. Yang: Conception or design of the work, Acquisition, analysis, Interpretation of data, Writing - original draft, Writing - review & editing. Y.G. Cao: Conception or design of the work, Acquisition, analysis, Interpretation of data, Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgment
The authors approved the final version to be published.
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